Summary
An improvement of Harris' theorem on percolation is obtained; it implies relations between critical points of matching graphs of the type of the one stated by Essam and Sykes. As another consequence, it is proved that the percolation probability, as a function of the probability of occupation of a given site, is infinitely differentiable, except at most in the critical point.
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Russo, L. A note on percolation. Z. Wahrscheinlichkeitstheorie verw Gebiete 43, 39–48 (1978). https://doi.org/10.1007/BF00535274
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DOI: https://doi.org/10.1007/BF00535274